In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial...

of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter...

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other....

In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on...

{\displaystyle X(M)\to X(N),} are n-connected are said to satisfy a homotopy principle or "h-principle". There are a number of powerful general techniques for proving...

that this reduces to homotopy theory, and the homotopy principle gives general conditions and reasons for PDRs to reduce to homotopy theory. Immersed submanifold...

895842. S2CID 206590734.* 5-manifold Axiom A Geometric mechanics Homotopy principle Mean value problem Smale, Steve (1985). "On the Efficiency of Algorithms...

to always be immersions or not), which is an example of an h-principle (homotopy-principle), meaning that geometry reduces to topology. This fact (that...

This can then be used to prove the commutativity of the higher homotopy groups. The principle is named after Beno Eckmann and Peter Hilton, who used it in...

Forstnerič, Franc (2017). Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete...

Gromov, a prominent developer of geometric group theory, inventor of homotopy principle, introduced Gromov's compactness theorem, Gromov norm, Gromov product...

Gromov, a prominent developer of geometric group theory, inventor of homotopy principle, introduced Gromov's compactness theorems, Gromov norm, Gromov product...

algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category Ho ( Top ) {\displaystyle...

computational method used in numerical algebraic geometry is homotopy continuation, in which a homotopy is formed between two polynomial systems, and the isolated...

equivalence of categories Weak equivalence (homotopy theory) Weak equivalence (formal languages) Weak equivalence principle This disambiguation page lists mathematics...

and contact structures. Eliashberg worked on various aspects of the h-principle, introduced by Mikhail Gromov, and he wrote in 2002 an introductory book...

Homology is a topological invariant, and moreover a homotopy invariant: Two topological spaces that are homotopy equivalent have isomorphic homology groups. It...

the first time. See h-principle for further generalizations. Smale's original proof was indirect: he identified (regular homotopy) classes of immersions...

Gromov, a prominent developer of geometric group theory, inventor of homotopy principle, introduced Gromov's compactness theorems in geometry and topology...

theorem Chern's conjecture (affine geometry) Differential structure Homotopy principle Immersion (mathematics) Whitney embedding theorem The Cr section theorem...

univalent foundations and related to it homotopy type theory. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties...

where d ≥ 3, then this homotopy class only has one element. If M is ⁠ R 2 {\displaystyle \mathbb {R} ^{2}} ⁠, then this homotopy class has countably many...

theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups Periodic function, a function whose output contains...

circle. The set of homotopy classes of maps from a circle to a topological space form a group, which is called the first homotopy group or fundamental...

Kirszenblat and J. Hyam Rubinstein. A proof characterizing Dubins paths in homotopy classes has been given by J. Ayala. The Dubins path is commonly used in...

invertible complex matrices, which have the homotopy type of a circle. From the perspective of homotopy theory, a real line bundle therefore behaves...

the extension of order-theoretic duality to Boolean algebras S-duality (homotopy theory) List of dualities § Mathematics Dualistic cosmology, a twofold...

for homotopy groups of spheres. Computational methods for solving systems of polynomial equations. Brown has an algorithm to compute the homotopy groups...

some weakly contractible space, e.g., a topological space with vanishing homotopy groups. The classifying space has the property that any G principal bundle...

is an active area of research, one direction being the development of homotopy type theory. The first computer proof assistant, called Automath, used...